A002805 -id:A002805 - cp's OEIS frontend

A330734 a(n) = n - A309639(n), where A309639(n) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

0, 0, 0, 0, 0, 3, 0, 0, 0, 5, 0, 8, 0, 7, 10, 0, 0, 9, 0, 15, 12, 11, 0, 15, 0, 13, 0, 21, 0, 25, 0, 0, 22, 17, 28, 27, 0, 19, 26, 32, 0, 33, 0, 33, 36, 23, 0, 32, 0, 25, 34, 39, 0, 27, 44, 48, 38, 29, 0, 55, 0, 31, 54, 0, 52, 55, 0, 51, 45, 63, 0, 63, 0, 37, 50, 57, 66, 65, 0, 64, 0, 41, 0, 75, 68, 43, 58, 77, 0, 81

Offset: 1

Antti Karttunen, Jan 10 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961 (indices of zeros), A309639, A330691, A330692, A330735.

PARI
```
A330734(n) = (n-A309639(n));
```

a(n) = n - A309639(n).

A330736 Numbers k such that k is not a multiple of A309639(k), where A309639(k) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by k.

Original entry on oeis.org

21, 24, 42, 69, 84, 105, 115, 120, 138, 168, 171, 201, 207, 210, 225, 230, 276, 301, 329, 342, 345, 402, 407, 414, 420, 450, 451, 460, 473, 483, 505, 515, 602, 603, 605, 639, 658, 684, 690, 759, 804, 805, 814, 828, 840, 855, 869, 891, 897, 900, 902, 903, 913, 920, 946, 966, 987, 1005, 1010, 1030, 1035, 1173, 1197

Offset: 1

Antti Karttunen, Jan 10 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2366

Positions of nonzero terms in A330735.

Cf. A002805, A309639.

A349851 Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.

Original entry on oeis.org

8, 4, 6, 2, 9, 7, 2, 4, 9, 2, 9, 9, 9, 7, 1, 2, 2, 4, 5, 3, 9, 7, 7, 2, 5, 0, 5, 8, 2, 5, 5, 1, 1, 3, 6, 6, 2, 6, 9, 8, 7, 0, 7, 6, 3, 1, 5, 6, 4, 4, 2, 8, 0, 7, 2, 2, 9, 4, 1, 4, 1, 0, 9, 6, 8, 8, 5, 9, 7, 3, 8, 8, 6, 4, 2, 9, 4, 8, 7, 9, 0, 7, 2, 5, 0, 0, 8, 2, 6, 0, 8, 9, 5, 0, 7, 1, 1, 6, 7, 9, 3, 1, 5, 3, 1

Offset: 1

Amiram Eldar, Dec 02 2021

			8.46297249299971224539772505825511366269870763156442...

Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

Cf. A000032, A001008, A001622, A002162, A002390, A002805, A349850.

RealDigits[6*Log[2] + 4*Sqrt[5]*Log[GoldenRatio], 10, 100][[1]]

Equals log(64*phi^(4*sqrt(5))) = 6*log(2) + 4*sqrt(5)*log(phi), where phi is the golden ratio (A001622).

A035047 Denominators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

Original entry on oeis.org

1, 2, 3, 4, 15, 12, 105, 24, 315, 120, 3465, 40, 45045, 280, 45045, 560, 765765, 5040, 14549535, 5040, 14549535, 55440, 334639305, 55440, 1673196525, 720720, 5019589575, 720720, 145568097675, 720720

Offset: 1

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Transforms

Cf. A035048.

S:= series(log(1-x)/(x^2-1), x, 101):
seq(denom(coeff(S,x,j)),j=1..100); # Robert Israel, Jun 02 2015

a(n)=denominator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003

a(n) = denominator((-1)^(n+1)*1/2*(log(2)+(-1)^(n+1)*(gamma+1/2*(psi(1+n/2)-psi(3/2+n/2))+psi(2+n)))), with gamma the Euler-Mascheroni constant. - [Gerry Martens, Apr 28 2011]

A065454 Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).

Original entry on oeis.org

9, 11, 13, 14, 21, 25, 27, 29, 33, 34, 35, 37, 38, 39, 44, 45, 47, 49, 50, 51, 54, 55, 56, 57, 59, 61, 64, 67, 69, 73, 74, 75, 77, 79, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 105, 107, 110, 111, 113, 114, 115, 116, 117, 118, 121, 122, 123, 125

Offset: 1

Benoit Cloitre, Nov 24 2001

nonn

Shiu (2016) proved that this sequence is infinite. Wu and Chen (2019) proved that the asymptotic density of this sequence is 1. - Amiram Eldar, Jan 29 2021

			For example: H(11) = 83711/27720, H(12) = 86021/27720 and so a(2) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
Bing-Ling Wu and Yong-Gao Chen, On the denominators of harmonic numbers, II, Journal of Number Theory, Vol. 200 (2019), pp. 397-406.

Cf. A001008, A002805.

Position[Partition[Denominator @ HarmonicNumber[Range[126]], 2, 1], {x_, x_}] // Flatten (* Amiram Eldar, Jan 29 2021 *)

A103931 Denominators of squares of harmonic numbers A001008/A002805.

Original entry on oeis.org

1, 4, 36, 144, 3600, 400, 19600, 78400, 6350400, 6350400, 768398400, 768398400, 129859329600, 129859329600, 129859329600, 519437318400, 150117385017600, 16679709446400, 6021375110150400, 240855004406016, 26761667156224

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding numerators are given in A103930. For the rational see the link there.

Denominator[HarmonicNumber[Range[30]]^2] (* Harvey P. Dale, Oct 16 2022 *)

a(n)=denominator(H(n)^2), with the harmonic numbers H(n)=A001008(n)/A002805(n), n>=1.

A103933 Denominators of first difference of squares of harmonic numbers A001008/A002805.

Original entry on oeis.org

1, 4, 9, 48, 150, 90, 490, 2240, 11340, 2520, 152460, 83160, 2342340, 2522520, 540540, 11531520, 104144040, 110270160, 737176440, 775975200, 162954792, 56904848, 1368302936, 2141691552, 111546435000, 116008292400, 1084231348200

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding numerators are given in A103932. For the rationals see the link there.

Denominator[Differences[HarmonicNumber[Range[0,30]]^2]] (* Harvey P. Dale, Sep 09 2012 *)

a(n)=numerator(r(n)), with the rationals r(n)=H(n)^2-H(n-1)^2 where H(n)= A001008(n)/A002805(n), n>=1, H(0):=0.

Offset corrected by Mohammed Yaseen, Aug 09 2023

A329061 Greatest k such that A002805(k) is not divisible by n, or a(n) = 0 if there's no such k.

Original entry on oeis.org

0, 1, 68, 3, 124, 68, 719102, 7, 206, 124, 11130347490407364042652446389727, 68, 2196, 719102, 124, 15, 4912, 206, 16128612858, 124, 719102, 11130347490407364042652446389727, 12166, 68, 624, 2196, 620, 719102, 20171036, 124, 27488495831, 31, 11130347490407364042652446389727

Offset: 1

Jinyuan Wang, Dec 07 2019

nonn

There are two cases where a(n) = 0: (a) n divides A002805(k) for all k, which only happens for n = 1; (b) there are infinitely many k such that n does not divide A002805(k), which may happen for some primes p and their multiples.
For k > a(n) > 0, A002805(k) is always divisible by n.
For prime p and k >= p, A002805(k) = (the denominator of s + (Sum_{i=1..floor(k/p)} 1/i)/p) is not divisible by p if and only if p divides A001008(floor(k/p)) = (the numerator of Sum_{i=1..floor(k/p)} 1/i), because the denominator of s = Sum_{1 <= i <= k, i is not divisible by p} 1/i can never be divisible by p.
If k == -1 or 0 (mod p), then p divides A001008(k) iff p^2 divides A001008(floor(k/p)), otherwise p divides A001008(k) iff p divides the numerator of (Sum_{i=floor(k/p)*p+1..k} 1/i) + (Sum_{i=1..floor(k/p)} 1/i)/p, where p is an odd prime and k >= p. (Since Sum_{i=1..p-1} (p-1)!/i = (-1)^((p-1)/2)*((p-1)/2)!*(Sum_{i=1..(p-1)/2} ((p-1)/2)!/i) + ((p-1)/2)!*(Sum_{i=1..(p-1)/2} (-1)^((p-1)/2)*((p-1)/2)!/(-i)) == 0 (mod p), odd prime p divides the numerator of Sum_{1 <= i <= floor(k/p)*p, i is not divisible by p} 1/i.)

			For p = 3, 3 divides numerator(1+1/2), so 2*3, 2*3 + 1 and 2*3 + 2 are such k that A002805(k) can't be divisible by 3. Similarly, 7*3, 7*3 + 1 and 7*3 + 2 are such k. Mod(A001008(7), 3) > 0 and Mod(numerator(1/22 + (Sum_{i=1..7} 1/i)/3), 3) = 0, hence 3 divides A001008(22), which means 22*3, 22*3 + 1 and 22*3 + 2 are also such k. a(3) = 68 because A001008(k) can never be divisible by 3 for k = 66, 67 and 68.

Jinyuan Wang, PARI program and examples for n <= 10
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A177734, A329293.

If n = Product_{j=1..i} p_j^e_j, p_1 < ... < p_i are primes and a(p_j^e_j) > 0, then a(n) = Max_{j=1..i} a(p_j^e_j).
a(p^e) = p^(e-1)*(a(p)+1) - 1 for prime p and a(p) > 0. Proof: A001008(k)/A002805(k) = (Sum_{1 <= i <= k, i is not divisible by p^e} 1/i) + (Sum_{i=1..floor(k/p^e)} 1/i)/p^e), hence A002805(k) is not divisible by p^e if and only if p divides A001008(floor(k/p^e)). From the comment, we know that (a(p)+1)/p - 1 is the greatest m such that p divides A001008(m). Therefore, a(p^e) = p^e*((a(p)+1)/p-1) + p^e - 1 = p^(e-1)*(a(p)+1) - 1.
a(prime(i)) = (A177734(i)+1)*prime(i) - 1, where prime(i) = A000040(i). - Jinyuan Wang, Feb 06 2020

More terms from Jinyuan Wang, Feb 06 2020

A329293 Number of positive integers k such that A002805(k) is not divisible by n, or a(n) = 0 if there are infinitely many such numbers.

Original entry on oeis.org

0, 1, 11, 3, 19, 11, 97, 7, 35, 19

Offset: 1

Jinyuan Wang, Dec 27 2019

There are two cases where a(n) = 0: (a) n divides A002805(k) for all k, which only happens for n = 1; (b) there are infinitely many k such that n does not divide A002805(k), which may happen for some primes p and their multiples.
For prime p and k >= p, A002805(k) is not divisible by p if and only if p divides A001008(floor(k/p)), which means a(p) mod p = p - 1.
If k == -1 or 0 (mod p), then p divides A001008(k) iff p^2 divides A001008(floor(k/p)), otherwise p divides A001008(k) iff p divides the numerator of (Sum_{i=floor(k/p)*p+1..k} 1/i) + (Sum_{i=1..floor(k/p)} 1/i)/p, where p is an odd prime and k >= p. See A329061 for more information.

Jinyuan Wang, PARI program and examples for n <= 10
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A329061.

A331777 Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

Original entry on oeis.org

1, 1, 1, 0, -1, 1, -1, -43, 1831, 949, -137309, -85511, 3404045159, 777985057, -21024051077, -2192231411, 467347169033357, 10187765700589, -11741590582705819219, -3086703970985605357, 169597995722575162268081, 19606186988235984155519, -62715098968866173387571821

Offset: 0

N. J. A. Sloane, Feb 09 2020

Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, 40:2 (2016), 279-290. See Eq. (2.11).

Denominators are in A331778.

Cf. A001008, A001620, A002805, A093334, A238813, A331943.

Numerator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)

Sign of a(7) corrected and more terms from Vaclav Kotesovec, Feb 10 2020

cp's OEIS Frontend

A330734 a(n) = n - A309639(n), where A309639(n) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

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A330736 Numbers k such that k is not a multiple of A309639(k), where A309639(k) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by k.

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A349851 Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.

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A035047 Denominators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

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Maple

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A065454 Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).

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Mathematica

A103931 Denominators of squares of harmonic numbers A001008/A002805.

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Mathematica

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A103933 Denominators of first difference of squares of harmonic numbers A001008/A002805.

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Mathematica

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Extensions

A329061 Greatest k such that A002805(k) is not divisible by n, or a(n) = 0 if there's no such k.

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A329293 Number of positive integers k such that A002805(k) is not divisible by n, or a(n) = 0 if there are infinitely many such numbers.

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